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Simplifying 7n2 + 12n + -3 = 0 Reorder the terms: -3 + 12n + 7n2 = 0 Solving -3 + 12n + 7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.4285714286 + 1.714285714n + n2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + 1.714285714n + 0.4285714286 + n2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + 1.714285714n + n2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + 1.714285714n + n2 = 0 + 0.4285714286 1.714285714n + n2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 1.714285714n + n2 = 0.4285714286 The n term is 1.714285714n. Take half its coefficient (0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. 1.714285714n + 0.7346938773 + n2 = 0.4285714286 + 0.7346938773 Reorder the terms: 0.7346938773 + 1.714285714n + n2 = 0.4285714286 + 0.7346938773 Combine like terms: 0.4285714286 + 0.7346938773 = 1.1632653059 0.7346938773 + 1.714285714n + n2 = 1.1632653059 Factor a perfect square on the left side: (n + 0.857142857)(n + 0.857142857) = 1.1632653059 Calculate the square root of the right side: 1.078547776 Break this problem into two subproblems by setting (n + 0.857142857) equal to 1.078547776 and -1.078547776.Subproblem 1
n + 0.857142857 = 1.078547776 Simplifying n + 0.857142857 = 1.078547776 Reorder the terms: 0.857142857 + n = 1.078547776 Solving 0.857142857 + n = 1.078547776 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = 1.078547776 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = 1.078547776 + -0.857142857 n = 1.078547776 + -0.857142857 Combine like terms: 1.078547776 + -0.857142857 = 0.221404919 n = 0.221404919 Simplifying n = 0.221404919Subproblem 2
n + 0.857142857 = -1.078547776 Simplifying n + 0.857142857 = -1.078547776 Reorder the terms: 0.857142857 + n = -1.078547776 Solving 0.857142857 + n = -1.078547776 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = -1.078547776 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = -1.078547776 + -0.857142857 n = -1.078547776 + -0.857142857 Combine like terms: -1.078547776 + -0.857142857 = -1.935690633 n = -1.935690633 Simplifying n = -1.935690633Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.221404919, -1.935690633}
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